by Niven's theorem a rational cosine coincides with a rational angle if and only if the cosine equals 0, ±1/2, or ±1. The only ones of these giving an angle strictly between 0° and 180° are the cosine value 1/2 with the angle 60°, the cosine value –1/2 with the angle 120°, and the cosine value 0 with the angle 90°. The only combination of three of these, allowing multiple use of any of them and summing to 180°, is three 60° angles. Integer triangles with integer ratio of circumradius to inradius Conditions are known in terms of elliptic curves for an integer triangle to have an integer ratio N of the circumradius to the inradius.Goehl, John F. Jr., "More integer triangles with R/r = N", Forum Geometricorum 12, 2012: pp. 27−28 The smallest case, that of the equilateral triangle, has N = 2. In every known case, N ≡ 2 (mod 8)—that is, N – 2 is divisible by 8. 5-Con triangle pairs A 5-Con triangle pair is a pair of triangles that are similar but not congruent and that share three angles and two sidelengths. Primitive integer 5-Con triangles, in which the four distinct integer sides (two sides each appearing in both triangles, and one other side in each triangle) share no prime factor, have triples of sides and for positive coprime integers x and y. The smallest example is the pair (8, 12, 18), (12, 18, 27), generated by x = 2, y'' = 3. Particular integer triangles The only triangle with consecutive integers for sides and area has sides (3, 4, 5) and area 6. The only triangle with consecutive integers for an altitude and the sides has sides (13, 14, 15) and altitude from side 14 equal to 12. The (2, 3, 4) triangle and its multiples are the only triangles with integer sides in arithmetic progression and having the complementary exterior angle property. This property states that if angle C is obtuse and if a segment is dropped from B meeting perpendicularly AC extended at P, then ∠CAB=2∠CBP. The (3, 4, 5) triangle and its multiples are the only integer right triangles having sides in arithmetic progression. The (4, 5, 6) triangle and its multiples are the only triangles with one angle being twice another and having integer sides in arithmetic progression. The (3, 5, 7) triangle and its multiples are the only triangles